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1. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor; Zumbrun, Kevin (January 2022, Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$$ (0, +\infty) $$\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $$\end{document} stable manifolds of such equations, showing that \begin{document}$$ L^2_{loc} $$\end{document} solutions that remain sufficiently small in \begin{document}$$ L^\infty $$\end{document} (i) decay exponentially, and (ii) are \begin{document}$$ C^\infty $$\end{document} for \begin{document}$$ t>0 $$\end{document}, hence lie eventually in the \begin{document}$$ H^1 $$\end{document}$ stable manifold constructed by Pogan and Zumbrun. 

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2. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan; Zhang, Yanzhi (January 2022, Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$$ (- \Delta)^\frac{{ \alpha}}{{2}} $$\end{document} for \begin{document}$$ \alpha \in (0, 2) $$\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^2) $$\end{document}, while \begin{document}$$ {\mathcal O}(h^4) $$\end{document} for quadratic basis functions with \begin{document}$ h $$\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$$ \alpha \in (0, 2) $$\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$$ u \in C^{m, l}(\bar{ \Omega}) $$\end{document} for \begin{document}$$ m \in {\mathbb N} $$\end{document} and \begin{document}$$ 0 < l < 1 $$\end{document}, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $$\end{document} for constant and linear basis functions, while \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $$\end{document}$ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency. 

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3. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane; Demers, Mark F. (January 2022, Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $$\end{document} on the two-torus, we find \begin{document}$$ t_*>1 $$\end{document} such that for each \begin{document}$$ t\in (0,t_*) $$\end{document} there exists a unique equilibrium state \begin{document}$$ \mu_t $$\end{document} for \begin{document}$$ - t\log J^uT $$\end{document}, and \begin{document}$$ \mu_t $$\end{document} is \begin{document}$$ T $$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$$ - \log J^uT $$\end{document}.) We show that \begin{document}$$ \mu_t $$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $$\end{document} is analytic on \begin{document}$$ (0,t_*) $$\end{document}. In addition, \begin{document}$$ P(t) $$\end{document} is strictly convex if and only if \begin{document}$$ \log J^uT $$\end{document} is not \begin{document}$$ \mu_t $$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$$ t_a\ne t_b $$\end{document} with \begin{document}$$ \mu_{t_a} = \mu_{t_b} $$\end{document}, then \begin{document}$$ P(t) $$\end{document} is affine on \begin{document}$$ (0,t_*) $$\end{document}. An additional sparse recurrence condition gives \begin{document}$$ \lim_{t\downarrow 0} P(t) = P(0) $$\end{document}$. 

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4. Realizing arbitrary $$d$$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

   https://doi.org/10.3934/dcds.2021129  

   Fayad, Bassam; Saprykina, Maria (January 2022, Discrete & Continuous Dynamical Systems)

   Any \begin{document}$ C^d $$\end{document} conservative map \begin{document}$$ f $$\end{document} of the \begin{document}$$ d $$\end{document}-dimensional unit ball \begin{document}$$ {\mathbb B}^d $$\end{document}, \begin{document}$$ d\geq 2 $$\end{document}, can be realized by renormalized iteration of a \begin{document}$$ C^d $$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$$ {\mathbb B}^d $$\end{document}, arbitrarily close to identity in the \begin{document}$$ C^d $$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$$ C^d $$\end{document} change of coordinates is exactly \begin{document}$$ f $$\end{document}$. 

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5. Existence of global weak solutions of $ p $-Navier-Stokes equations

   https://doi.org/10.3934/dcdsb.2021051  

   Liu, Jian-Guo; Zhang, Zhaoyun (January 2022, Discrete & Continuous Dynamical Systems - B)

   This paper investigates the global existence of weak solutions for the incompressible \begin{document}$ p $$\end{document}-Navier-Stokes equations in \begin{document}$$ \mathbb{R}^d $$\end{document} \begin{document}$$ (2\leq d\leq p) $$\end{document}. The \begin{document}$$ p $$\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$$ p $$\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$$ p $$\end{document}-Laplacian of velocity and the \begin{document}$$ p $$\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$$ p $$\end{document}$ distances with constraint density to be characteristic functions. 

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